Wikipedia:Peer review/Manifold/archive1

manifold

This is an important concept in mathematics with many incarnations. It allows for a simple geometric interpretation of gluing paper together. It has been attempted to make this article accessible to non-mathematicians and reserve the technical details mostly for specific incarnations such as differentiable manifold and topological manifold. Therefore I would be very interested in finding out whether we succeeded in this at all. All other comments are also welcome of course. --MarSch 11:26, 20 October 2005 (UTC)[reply]

Cdc

Wow, very well done. I really appreciate the work done to make this comprehensible to non-math people (like me). A few comments, from someone who had absolutely no idea what a manifold was before reading the article:

Looking only at the intro paragraphs: Manifold and Surface are be defined in terms of each other; a manifold is a generalized surface, and a surface is a 2-d manifold. Is there a way to describe one or both in some other terms, so I can get a clear picture of what's going on from the first paragraphs?
I've rewritten the intro to clear this up. --MarSch 11:48, 23 October 2005 (UTC)[reply]
I feel like there's some specific usage of the term simple here, but I'm not sure. Things can be simple in many ways; easy to understand, having few parts, etc. Since simplicity seems to be an important attribute of manifolds, would it help to define more specifically what the term means here?
I've rewritten the intro to clear this up. --MarSch 11:48, 23 October 2005 (UTC)[reply]
I don't usually use "you" and "we" when writing Wikipedia articles - it sounds somehow too informal to me. I'm aware that mathematics has its own writing style, so maybe it's okay here, but it's something to consider.
I will try to remove this writing style. --MarSch 11:48, 23 October 2005 (UTC)[reply]
I didn't get why the Hausdorff assumption is important enough for this general article - needs more context, maybe?
it is not important and way too technical and I've moved it to the more specialized topological manifold.--MarSch 10:42, 23 October 2005 (UTC)[reply]
Are there any applications or implications of this that can be briefly mentioned? Are these used in engineering or modeling or anything?
I've rewritten the intro to clear this up. --MarSch 11:48, 23 October 2005 (UTC)[reply]

This is a nice piece of work! CDC (talk) 22:22, 21 October 2005 (UTC)[reply]

Thanks for your review. --MarSch 11:48, 23 October 2005 (UTC)[reply]

Tony1

I like this a lot, but it needs a few changes.
- The lead is unsatisfactory. First sentence too technical—there's info further down that is more accessible to non-specialists, so why not present an opening definition that will be more widely comprehensible? Remove 'technically'? 'Much of the terminology is inspired'—insert 'connected with manifolds' after 'terminology'? 'In the remainder of this article we will give'—better as 'The remainder of this article will provide ...'. Three overly short paragraphs. Needs flow and needs to engage your readers.
- 'A manifold is a space that looks, when examined close up, like'—clumsy word order. Same for 'and is, in this sense, like a'—so check through the whole text for dependent clauses that are embedded in awkward positions.
  - I've rewritten the intro and it avoids this now --MarSch 11:56, 23 October 2005 (UTC)[reply]
- 'compared to' for similarities; 'compared with' for differences.
  - fixed --MarSch 11:56, 23 October 2005 (UTC)[reply]
'Such a'—better nowadays as 'This'.
- One example is given of a specific notion, but there are many more examples. Therefore "This" would not be correct and neither would "These". Perhaps this is alright after all? --MarSch 11:56, 23 October 2005 (UTC)[reply]
Italics for 'Figure 1' etc, is harder to read and unnecessary. Better to reserve that highlighting for technical terms.
- fixed --MarSch 11:59, 23 October 2005 (UTC)[reply]
Clean up the formatting of the reference list. Use all initials rather than first names? Punctuation needs to be consistent. 'provides', not 'gives'; 'in undergraduate ...' Hypy comment for Milnor. 'The 1851 doctoral thesis in which "manifold" (Mannigfaltigkeit) first appears.'—How many 1851 doctorates were there?
- I kind of like to keep the first names. I've fixed the punctuation and grammar/style errors. What is "Hypy"? There were probably many 1851 doctoral theses, but only one by Milnor. --MarSch 12:12, 23 October 2005 (UTC)[reply]
  - Agree that "best math book ever written" is hype (though I like the book). The doctoral thesis is by Riemann (not Milnor); those interested in history care about the date and the context (presumably). --KSmrq^T 22:36, 23 October 2005 (UTC)[reply]
The images are FUNKY!
- Please explain this more. I'm not even sure wether this means bad or is a compliment, though I'm assuming the first.--MarSch 12:12, 23 October 2005 (UTC)[reply]

Tony 02:49, 22 October 2005 (UTC)[reply]

Dethomas

I agree, this is a well done article. I like the second and third person here, the you's and we's, they help to make the material more accessible to the casual reader.

The pictures rock, but the article needs a hook in the lead paragraph. I left out the wiki formatting and links, but how about something like this for a lead?

"While many people think of x's and y's and long schoolroom afternoons when they think of mathematics, many mathematicians think about shapes and surfaces. But our normal three dimensional space of width, height and depth is often inadequate for reasoning about mathematical problems. Over time, a variety of ideas have converged on the idea of a manifold as a way to help think about surfaces.

An everyday example of this sort of thinking is the common street map, which uses a two dimensional drawing to represent features of the earth, whose surface is three dimensional. Indeed, much of the terminology of manifolds is inspired by map-making or cartography, we speak of an atlas of charts which can be pieced together as a patchwork to describe a manifold." (by User:Dethomas --MarSch 10:27, 23 October 2005 (UTC))[reply]

Kill the opening and start from 'Our normal three-dimensional perception of ....' Tony 11:31, 23 October 2005 (UTC)[reply]

Thanks for your suggestion. Actually we like to say that the surface of the Earth is 2 dimensional. Also manifolds are about much more than only surfaces. I hope you like better the new intro I've created. --MarSch 12:20, 23 October 2005 (UTC)[reply]

The current opening paragraph doesn't engage the casual reader, while the second paragraph,if he gets that far, dumps a load of complexity on his head. The first paragraph of the Introduction is a much better answer to the reader's implicit "Why are manifolds important?" question.

Any lead paragraph needs to tell the reader why the material is worth his time. So what are manifolds and why should the reader care? Because they aid in mathematical reasoning about surfaces, shapes and spaces in ways that everyday three space is ill-suited to do. Let's just say so. I'm quite willing to let the "long schoolroom afternoons" go, but most readers need a boost to get over the "eek, math!!" threshold. I think the street map analogy or something of that nature places the reader in familiar territory (as do the words "chart" and "atlas") while opening the door to the depth of detail in the article. But that's just my opinion.

Dethomas 00:24, 24 October 2005 (UTC)[reply]

The closest I can get to a street map example is the example of charts in an atlas of Earth. This is what I wrote about it [1] a while ago which later got ruthlessly cut [2]. This is what I wrote

Imagine you have a few sheets of paper and some glue. The paper is of a special high-quality kind that can be strechted and molded into whatever shape you want and it never tears. You could cover the Earth with just two such sheets. One strechted over the North pole all the way down to Antarctica and another stretched over the South pole all the way up to Greenland, with a bit of glue at the tropics where they overlap. You have just proved that the surface of the Earth is a paper manifold!

Is this like what you have in mind? --MarSch 16:22, 28 October 2005 (UTC)[reply]

Not really, not for a lead or opening paragraph. A good lead convinces the reader, in a few sentences, that the material is worth his time and effort. My sense is you are being too literal (pedantic?) with the "earth's surface is two dimensional" mathematical idea, and hence missing the point. To the the target audience, the non-technical reader, the idea that the earth's surface has height, width, and depth but we commonly represent it with flat, two dimensional chucks of paper called "maps" is understandable and nonthreatening. By analogy, we can proceed from the familiar concept of a street map to the unfamiliar concept of a manifold, draw the reader into the article, and let the Introduction section do it's job.

The current (Oct 29) lead is too big, too wooden and does little to draw the non-technical reader into the body of the article. So I'm still suggesting something like this:

"Our mundane notions of width, height and depth are often inadequate for reasoning about some types of mathematical problems. Over time, a variety of ideas have converged on the idea of a manifold as a way to help mathematicians think about surfaces and related topics. An everyday example of this sort of thinking is the common street map, which uses a two dimensional drawing to represent features of the earth, whose surface is actually three dimensional on the scale of daily experience. Indeed, much of the terminology of manifolds is inspired by map-making or cartography. We speak of an atlas of charts, which can be pieced together as a [[[patchwork]]] to describe a manifold."

You can take this verbatim as the lead, or you can bless it and I'll put it in, or somebody can write a new lead, but if we are trying to reach non-technical readers, we should ditch the current opening paragraphs if favor of something Joe Everyman can read without passing out.

Dethomas 05:39, 30 October 2005 (UTC)[reply]

Unfortunately I cannot bless it. Projecting out the height to construct a city map has nothing to do with manifolds. Or perhaps I should say that this is way too trivial to mention in this article. Everybody knows how a chart can represent a piece of the earth. Stating that the surface of the Earth is 3D is unacceptable too. What manifolds are about for example is that even though you cannot make a chart which covers the Earth, you can nevertheless bring a lot of mathematics from the plane over to the sphere.

It would be more helpful if you tried to make specific remarks about the intro, so that I can fix/rewrite things.--MarSch 12:59, 31 October 2005 (UTC)[reply]

I thought I did make specific remarks, and offered specific remedies to what I saw as essential problems. But that's just an opinion.

In any event, an effective lead paragraph draws in the casual reader. If a trivial example does the job, use it. I think it would be more helpful if you remembered the article is aimed at a non-technical audience, that the rigor of a textbook is inappropriate, and give me the benefit of the doubt, as opposed to disrespect of your contempt.

The phrase "you can nevertheless bring a lot of mathematics from the plane over to the sphere' is precisely what you what to tell the reader in a lead, and precisely what my examples have suggested. Read what's in the comment, not what you wish was in the comment.

Or not. Your burden.

Dethomas 18:05, 1 November 2005 (UTC)[reply]

I didn't mean to disrespect you or show contempt. If I have I apologize. Unfortunately that doesn't change the fact that the lead you suggested is unacceptable for the reasons I stated. I know you have said that the lead should draw in the audience and I agree. I just don't know what will do that for laymen. To me the mention of usability for general relativity is enough to get me interested. On the other hand I consider "you can nevertheless bring a lot of mathematics from the plane over to the sphere" only of superficial interest. So how can a general audience appreciate the content of that statement? Apparently I'm wrong. So thanks for that. I'll incorporate this in the lead ASAP. When I said specific I meant extremely specific. Tell me what you think of each sentence, whether it is clear or interesting. Questions which are unanswered. Thanks in advance. --MarSch 11:20, 3 November 2005 (UTC)[reply]

KSmrq

Good to see this moving forward. A number of goals agreed in the talk archives still could use some work. One lingering concern is the lack of a link, say in Other types and generalizations of manifolds, to the original and still important type, the Riemannian manifold. --KSmrq^T 22:36, 23 October 2005 (UTC)[reply]

Riemannian manifold-link is buried inside differentiable manifold which is linked to. Perhaps the history needs to be augmented? --MarSch 16:12, 28 October 2005 (UTC)[reply]

Exactly what goals do you think need work?--MarSch 16:15, 28 October 2005 (UTC)[reply]

NatusRoma

My comments hardly deserve their own subsubheading, but that seems to be the way. I suggest moving some of the technical details in the introduction into paragraphs in the lead. As a mathematician, I wanted to engage the mathematics more quickly, and the lead doesn't really allow me to do that. In fact, I'll go so far as to say that there should surely be technical details in the first sentence of the article. It's an article about mathematics, so the content should be about mathematics. NatusRoma 05:41, 3 November 2005 (UTC)[reply]

We decided to make manifold a general article and to keep it free of technical details. Those are to be addressed in topological manifold and differentiable manifold and their cousins. Perhaps this fact should be stated before the lead. Does this address your concern? Which things from the intro do you think should be moved to the lead? --MarSch 11:25, 3 November 2005 (UTC)[reply]

Ah, I hadn't noticed that. My concerns are alleviated, then. NatusRoma 19:31, 12 November 2005 (UTC)[reply]

Deryck

I personally think, as the first leading paragraph stands for an introduction, it might be better to merge the "introduction" section into up above the TOC. Other things are pretty good. However, I don't know if it can go through in case you put it up for FAC -- not everybody's interested in this difficult mathematical thing. Deryc k C. 08:14, 9 November 2005 (UTC)[reply]

I agree with merging the lead and intro above the TOC. Unfortunately many contributors feel it is paramount that the lead not be longer than the few sentences it is now :( . Nobody is interested in every article. However if this article is too difficult please specify which bits are, so they can be fixed if possible. --MarSch 14:13, 17 November 2005 (UTC)[reply]

Vb

YES. You MUST pu this on FAC! This is basically more interesting than shoe polish! More seriously, some comments:

"A manifold is a space that looks, locally, like a Euclidean space" is in contradistinction with "There are many different classes of manifolds. The simplest are topological manifolds, which look locally like some Euclidean space. Other classes of manifolds have additional structure. One of the most important kind of manifold is the differentiable manifold, which has a structure that permits the application of calculus." Moreover I don't understand what is meant by "calculus" in this case. Something more explicit would help.
- Good catch. I've removed the whole first paragraph as it was way too specific and uninteresting for this overview article. --MarSch 14:07, 17 November 2005 (UTC)[reply]
- The calculus is difficult to catch in one sentence. It means that there is the notion of differentiabillity. Thus given a function from a differentiable manifold to the reals (also a diff. manifold) you can say whether it is differentiable. This is not possible for a topological manifold which is not a diff manifold. Vectorfields are introduced as derivations on differentiable functions. One can embed the function into the dual vector space by the evaluation. Elements of the dual vector space are called 1-forms. There is a notion of integration of 1-forms on diff. manifolds. Perhaps this can be boiled down to something about differentiabillity and integration. --MarSch 14:07, 17 November 2005 (UTC)[reply]
  - Yes I think you are right. It should be possible to replace calculus by a reference to integration and differentiation. Vb 16:45, 17 November 2005 (UTC)[reply]
Could you link "holistically" to something understandable or (better) explain it more simply. I looked in my English-French dictionary and "holistic" is translated aby "holistique" which doesn't help me.
- Have linked to holism. The opposite of reductionism. The resulting topology after gluing cannot be traced back to any of the constituent parts. All patches have trivial topology.--MarSch 14:35, 17 November 2005 (UTC)[reply]
  - Shouldn't the term holism be explained in two words in order to prevent clicking and flow break. Vb 16:45, 17 November 2005 (UTC)[reply]
I wonder whether Fig.1 or a simplified version of it could not be moved into the lead to make the article more appealing.
- I think fig1 is exactly where it should be, but perhaps a picture of a few manifolds could be moved to the lead. Perhaps a circle, a 2-sphere and a 2-torus?--MarSch 14:35, 17 November 2005 (UTC)[reply]
  - I think a circle divided into 2 arcs like describe in the lead would be good. However I wonder whether a map of the world wouldn't be better?
The sentence "The top, bottom, left, and right charts demonstrate that the circle is a manifold, but are not the only possible atlas." is clearly not correct. Well I guess.
- The sentence is correct. Please explain why you would think it is not, so that this statement can be explained a bit more. --MarSch 14:35, 17 November 2005 (UTC)[reply]
  - Charts do not demonstrate anything. The fact that each parts of the circle can be mapped to them demonstrate that the circle is a manifold. Charts are not an atlas. A set of charts is an atlas. Vb 16:45, 17 November 2005 (UTC)[reply]
This is unclear to me : "Viewed through the eyes of calculus, the circle transition function T is simply a function between open intervals, so we know what it means for T to be differentiable."
- The "Viewed through the eyes of calculus" is new and perhaps confusing and I will remove it. What the last part of the sentence is getting at, is that if T had been a function from the topological space we have just glued together (a circle) to itself, then we wouldn't know what it would mean for T to be differentiable. Since it is simply a mapping of the unit interval to itself we do know what it means. --MarSch 14:35, 17 November 2005 (UTC)[reply]
This sentence " If two charts overlap, parts of them represent the same region of the manifold, just as a map of Europe and a map of Asia may both contain Moscow." is very nice but should maybe appear above when the concept of transition atlas was introduced on the motivating example.
- Actually I think that section is pretty bad and needs a complete rethink. It is quite misleading. I will merge this sentence somewhere --MarSch 14:52, 17 November 2005 (UTC)[reply]
In "Motivational example: the circle", the explicit definition of Xtop and Xright should be given so that the algebra leading to $T(a)=\chi _{\mathrm {right} }\left(\chi _{\mathrm {top} }^{-1}(a)\right)=\chi _{\mathrm {right} }\left(a,{\sqrt {1-a^{2}}}\right)={\sqrt {1-a^{2}}}.$ $T(a)=\chi _{\mathrm {right} }\left(\chi _{\mathrm {top} }^{-1}(a)\right)=\chi _{\mathrm {right} }\left(a,{\sqrt {1-a^{2}}}\right)={\sqrt {1-a^{2}}}.$ can be followed more easily.
- Will do. --MarSch 14:52, 17 November 2005 (UTC)[reply]
In section "Differentiable manifolds", sentence "In particular it is possible to apply "calculus" on a differentiable manifold." should be expanded in order to make the reader understand what is meant by "calculus".
Isn't there a way to make the sentence "locally looks like the quotients of some simple space (e.g. Euclidean space) by the actions of various finite groups." understandable?
History should be moved in the front because it is clearly understandable by the layman.
It is written in the lead "Applications of manifolds to physics include differentiable manifolds which serve as the phase space in classical mechanics and four-dimensional pseudo-Riemannian manifolds which are used to model spacetime in general relativity." Could you make a section "applications" with examples from those topics in order to make the article interesting to a broader audience. In Bernard Schutz' Geometrical Methods of Mathematical Physics, Cambridge University Press, 1980 ISBN 0-521-29887-3, you'll find many applications interesting for physicists.
- Now this part disappeared. Why? These two applications are clearly the most important ones. Even if they are not the only ones. Vb 16:45, 17 November 2005 (UTC)[reply]

Vb 15:01, 11 November 2005 (UTC)[reply]

- - You are correct, they are quite important. I really disliked the rotation group example, so I was in remove mode. At the time I also thought it a good idea not to mention any examples in the lead, so at least they wouldn't be arbitrary. And I was trying to keep the lead short, so as not to offend too many people. I guess I failed. Thanks for setting me straight. --MarSch 11:51, 18 November 2005 (UTC)[reply]